Question

a triangle has two sides of length 19 and 3. what is the largest possible whole - number length for the third side?

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\). Let the sides be \(a = 19\), \(b = 3\), and \(c\) be the third side. Then \(19+3>c\) (i.e., \(c < 22\)) and \(19 + c>3\) (always true for positive \(c\)) and \(3 + c>19\) (i.e., \(c>16\)).

Step2: Find the largest whole - number

Since \(16 < c < 22\), the largest whole - number value of \(c\) is \(21\).

Answer:

\(21\)